MinT - Best Known (166−18, 166, s)-Nets in Base 3

Best Known (166−18, 166, s)-Nets in Base 3

(166−18, 166, 177154)-Net over F₃ — Constructive and digital

Digital (148, 166, 177154)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (1, 10, 7)-net over F₃, using
  - net from sequence [i] based on digital (1, 6)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 1 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
2. digital (138, 156, 177147)-net over F₃, using
  - net defined by OOA [i] based on linear OOA(3¹⁵⁶, 177147, F₃, 18, 18) (dual of [(177147, 18), 3188490, 19]-NRT-code), using
    - OA 9-folding and stacking [i] based on linear OA(3¹⁵⁶, 1594323, F₃, 18) (dual of [1594323, 1594167, 19]-code), using
      - 1 times truncation [i] based on linear OA(3¹⁵⁷, 1594324, F₃, 19) (dual of [1594324, 1594167, 20]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 1594324Â | 3²⁶âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]

(166−18, 166, 531461)-Net over F₃ — Digital

Digital (148, 166, 531461)-net over F₃, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(3¹⁶⁶, 531461, F₃, 3, 18) (dual of [(531461, 3), 1594217, 19]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3¹⁶⁶, 1594383, F₃, 18) (dual of [1594383, 1594217, 19]-code), using
  - discarding factors / shortening the dual code based on linear OA(3¹⁶⁶, 1594385, F₃, 18) (dual of [1594385, 1594219, 19]-code), using
    - constructionÂ X applied to C([0,9]) âŠ‚ C([0,6]) [i] based on
      1. linear OA(3¹⁵⁷, 1594324, F₃, 19) (dual of [1594324, 1594167, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324Â | 3²⁶âˆ’1, defining interval IÂ =Â [0,9], and minimum distance dÂ â‰¥ |{−9,−8,…,9}|+1Â =Â 20 (BCH-bound) [i]
      2. linear OA(3¹⁰⁵, 1594324, F₃, 13) (dual of [1594324, 1594219, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 1594324Â | 3²⁶âˆ’1, defining interval IÂ =Â [0,6], and minimum distance dÂ â‰¥ |{−6,−5,…,6}|+1Â =Â 14 (BCH-bound) [i]
      3. linear OA(3⁹, 61, F₃, 4) (dual of [61, 52, 5]-code), using
        discarding factors / shortening the dual code based on linear OA(3⁹, 80, F₃, 4) (dual of [80, 71, 5]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 80Â = 3⁴âˆ’1, defining interval IÂ =Â [0,2], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 5 [i]

(166−18, 166, large)-Net in Base 3 — Upper bound on s

There is no (148, 166, large)-net in base 3, because

16 times m-reduction [i] would yield (148, 150, large)-net in base 3, but
- mutually orthogonal hypercube bound [i]